Exact two-time correlation and response functions of the 1D Glauber-Ising model on the finite-length semi-open chain

The one-dimensional Glauber-Ising model quenched to zero temperature is the canonical exactly solvable example of phase-ordering ageing. Exact single-time and two-time correlators are known for the spatially infinite chain and for the finite periodic chain, and recently the exact single-time correlation function was obtained for a finite semi-open chain, in which the leftmost spin is permanently pinned and the correlator is forced to vanish at the free end. The explicitly identified open step for this geometry is the two-time level: derive exact closed-form expressions for the two-time correlation functions $C(t,s;x,y)=\langle\sigma_x(t)\sigma_y(s)\rangle$ between two mobile spins after a quench from a disordered state to $T=0$ , for the conjugate response functions $R(t,s;x,y)=\delta\langle\sigma_x(t)\rangle/\delta h_y(s)$ , and characterize their dynamical finite-size scaling — the boundary-condition dependence of the scaling functions, the plateau of the two-time autocorrelator deep in the finite-size regime, and possible boundary modifications of the autocorrelation exponent $\lambda=1$ and the limit fluctuation-dissipation ratio $X_\infty=1/2$ . No published solution is known for this geometry; the problem is open.

Background

The Glauber-Ising model is the Ising model of spins $\sigma_n=\pm1$ on a chain, equipped with Glauber's stochastic single-spin-flip dynamics: each spin flips at a rate depending on its two neighbours and chosen so that detailed balance holds at temperature $T$ , with the total magnetization not conserved. In one dimension the critical temperature is $T_c=0$ , and a quench from a disordered initial state to $T=0$ produces phase-ordering (coarsening): domains of parallel spins grow to a typical size $\ell(t)\sim t^{1/z}$ with dynamical exponent $z=2$ , and the system never equilibrates. This is the simplest realization of physical ageing, whose three defining properties are slow (non-exponential) relaxation, the absence of time-translation invariance, and dynamical scaling.

Ageing is quantified by two-time quantities. The two-time correlation function $C(t,s;x,y)=\langle\sigma_x(t)\sigma_y(s)\rangle$ compares the spin at site $y$ at the waiting time $s$ with the spin at site $x$ at the later time $t>s$ , and the two-time response function $R(t,s;x,y)=\delta\langle\sigma_x(t)\rangle/\delta h_y(s)$ , evaluated at zero field, measures the linear reaction to a small magnetic field $h_y(s)$ applied to site $y$ at the earlier time. Their scaling forms define the autocorrelation exponent $\lambda$ (through $C(t,s)\sim s^{-b}(t/s)^{-\lambda/z}$ for large time ratio $t/s$ ) and, via the fluctuation-dissipation ratio $X(t,s)=TR/\partial_sC$ (the temperature factor being understood in the critical-quench limit $T\to T_c=0^+$ of this $T_c=0$ system), its non-equilibrium long-time limit value $X_\infty$ . For the spatially infinite chain all of these are known exactly: the two-time correlators and responses with $X_\infty=1/2$ (Godrèche and Luck 2000), general multispin two-time functions (Mayer and Sollich 2004), and the universality of the fluctuation-dissipation ratio within two dynamical universality classes set by the initial state (Henkel and Schütz 2004).

For finite systems, dynamical finite-size scaling adds the inverse length $1/N$ as a relevant variable, and the finite-size scaling functions depend on the boundary conditions. Recently this programme has become exact in one dimension: on a periodic chain of finite length $N$ , the exact single-time and two-time correlators after a $T=0$ quench are known (Henkel 2025), and the two-time autocorrelator saturates at a plateau of height $C_\infty^{(2)}$ deep in the finite-size regime, obeying the scaling laws $C_\infty^{(2)}\propto N^{-\lambda}$ at fixed waiting time $s$ and $C_\infty^{(2)}\propto s^{\lambda/z-b}$ at fixed $N$ (for the Ising case $b=0$ , and the 1D $T=0$ quench has $z=2$ , $\lambda=1$ ); these laws were derived from generalised time-translation-invariance — the postulate, obtained out of equilibrium from a change of representation of the Lie algebra generators of dynamical scale-invariance and time-translation invariance, from which the algebraic scaling forms of two-time correlators and responses follow (Henkel 2025, Nucl. Phys. B) — and are exploited numerically in the two-dimensional Ising model (Warkotsch, Henkel and Janke 2026). Exact response functions on finite chains, for either geometry, have not been published — only their scaling forms are predicted.

The source paper addresses a new geometry: the semi-open chain of $N$ sites, in which — following an idea in Glauber's original 1963 paper — the leftmost spin is permanently pinned (it never flips) while the right end is free, the correlator being forced to vanish there. The single-time correlator with the pinned spin, $C(t;x)=\langle\sigma_0\sigma_x(t)\rangle$ , obeys in the continuum limit a diffusion equation $\partial_tC=\frac{1}{2}\partial_x^2C$ with $C(t;0)=1$ and $C(t;N)=0$ (the diffusion constant is halved relative to the periodic chain because only one of the two spins moves), and Henkel (2026) solved it exactly, expressing the finite-size scaling function through the Jacobi theta function $\vartheta_3$ (a classical special function of two arguments, a real variable and a nome $q$ ) in the scaling variables $|x|/N$ and $t/N^2$ . By a duality, this correlator also gives the empty-interval probability — the probability that a given interval of consecutive sites contains no particles — of the one-dimensional coagulation-diffusion process ( $A+A\to A$ ) on a periodic finite ring. This was the first finite-size scaling study of time-space correlation functions in the Glauber-Ising model with free-type boundary conditions.

What the source leaves open is the two-time level. Its conclusions state that the exact single-time results 'can immediately be used as initial conditions for the calculation of two-time correlators and the exploration of novel finite-size effects therein', and a footnote concedes that correlators of two mobile spins — which two-time functions involve — 'would require to use correlators with two spatial variables ... beyond the scope of the present work'. The two-time correlation and response functions of the semi-open chain, and the ageing finite-size physics they encode (boundary-condition dependence of the scaling functions, plateau behaviour, possible boundary modifications of $\lambda$ and $X_\infty$ ), are therefore not known; deriving them is the present problem.

Problem Statement

Consider the one-dimensional Glauber-Ising model — Ising spins with Glauber's non-conserved single-spin-flip dynamics satisfying detailed balance — on a semi-open chain of $N$ sites, quenched at $t=0$ from a fully disordered initial state to $T=0$ , with the leftmost spin $\sigma_0$ permanently pinned (held fixed) and the right end free, so that in the continuum limit the single-time correlator with the pinned spin obeys $\partial_tC=\frac{1}{2}\partial_x^2C$ with $C(t;0)=1$ and $C(t;N)=0$ . For this geometry, quench and initial state:

(1) Derive exact closed-form expressions for the two-time correlation functions $C(t,s;x,y)=\langle\sigma_x(t)\sigma_y(s)\rangle,\qquad 0<x,y<N,$ between two mobile spins, including the two-time autocorrelation limits $C(t,s;x,x)$ . The exact single-time semi-open correlator of Henkel (2026) supplies the pinned-end boundary data $C(t,s;0,y)=C(s;y)$ ; the equal-time two-spin correlator $C(s,s;x,y)$ needed as the initial condition at $t=s$ may itself have to be derived exactly first.

(2) Derive the conjugate two-time response functions $R(t,s;x,y)=\left.\frac{\delta\langle\sigma_x(t)\rangle}{\delta h_y(s)}\right|_{h=0}$ to a small local magnetic field $h_y(s)$ , and the fluctuation-dissipation ratio $X(t,s)=TR/\partial_sC$ , with the temperature factor understood in the critical-quench limit $T\to T_c=0^+$ (exact response functions are unknown even on the periodic finite chain).

(3) Extract from these exact forms the dynamical finite-size scaling behaviour in the scaling variables $|x|/N$ , $|y|/N$ , $t/N^2$ , $s/N^2$ : the boundary-condition dependence of the scaling functions, by comparison with the known periodic-chain and infinite-chain exact results; the behaviour of the two-time autocorrelator deep in the finite-size regime, in particular whether it saturates at a plateau whose height obeys the same laws as on the periodic chain ( $C_\infty^{(2)}\propto N^{-\lambda}$ at fixed $s$ , $C_\infty^{(2)}\propto s^{\lambda/z-b}$ at fixed $N$ , with infinite-chain values $z=2$ , $\lambda=1$ , $b=0$ ); and whether proximity to the pinned or the free boundary modifies the autocorrelation exponent $\lambda=1$ or the limit fluctuation-dissipation ratio $X_\infty=1/2$ .

The three parts jointly constitute one objective — the exact ageing dynamics of this geometry. Equivalent formulations (for example via the dual one-dimensional coagulation-diffusion process on a periodic finite ring and its two-time empty-interval probabilities) and any derivation technique are admissible, provided the results are exact and refer to this geometry, quench protocol and initial state.

References

Malte Henkel, 'Exact solution of the Glauber-Ising model on the finite-length semi-open chain', Entropy 28 (2026) 771, doi:10.3390/e28070771, arXiv:2606.16848, https://arxiv.org/abs/2606.16848
Malte Henkel, 'Finite-size scaling in the ageing dynamics of the 1D Glauber-Ising model', Entropy 27 (2025) 139, doi:10.3390/e27020139, arXiv:2501.05912, https://arxiv.org/abs/2501.05912
Malte Henkel, 'Physical ageing from generalised time-translation-invariance', Nuclear Physics B 1017 (2025) 116968, doi:10.1016/j.nuclphysb.2025.116968, arXiv:2504.16857, https://arxiv.org/abs/2504.16857
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Malte Henkel and Gunter M. Schütz, 'On the universality of the fluctuation-dissipation ratio in non-equilibrium critical dynamics', Journal of Physics A 37 (2004) 591-604, doi:10.1088/0305-4470/37/3/004, arXiv:cond-mat/0308466, https://arxiv.org/abs/cond-mat/0308466
Xavier Durang, Jean-Yves Fortin and Malte Henkel, 'Exact two-time correlation and response functions in the one-dimensional coagulation-diffusion process by the empty-interval method', Journal of Statistical Mechanics (2011) P02030, doi:10.1088/1742-5468/2011/02/P02030, arXiv:1012.4724, https://arxiv.org/abs/1012.4724
Xavier Durang, Jean-Yves Fortin, Diego Del Biondo, Malte Henkel and Jean Richert, 'Exact correlations in the one-dimensional coagulation-diffusion process investigated by the empty-interval method', Journal of Statistical Mechanics (2010) P04002, doi:10.1088/1742-5468/2010/04/P04002, arXiv:1001.3526, https://arxiv.org/abs/1001.3526
Dustin Warkotsch, Malte Henkel and Wolfhard Janke, 'Harnessing finite-size effects to gauge aging in the 2D Ising model', Journal of Statistical Mechanics (2026) 073208, doi:10.1088/1742-5468/ae8439, arXiv:2607.11524, https://arxiv.org/abs/2607.11524
Roy J. Glauber, 'Time-dependent statistics of the Ising model', Journal of Mathematical Physics 4 (1963) 294-307, doi:10.1063/1.1703954, https://doi.org/10.1063/1.1703954
Malte Henkel and Stoimen Stoimenov, 'Correlators in phase-ordering from Schrödinger-invariance', Nuclear Physics B 1020 (2025) 117151, doi:10.1016/j.nuclphysb.2025.117151, arXiv:2508.08963, https://arxiv.org/abs/2508.08963